Devin H. answered 05/04/19
Tutor in various subjects, and general knowledge enthusiast.
"Vanity of vanities! All things are vanity." - Eccl. 1:2
The strongest connection between Ecclesiastes and the New Testament is that the New Testament, the Gospels in particular, seem to introduce an element that Qoheleth had not considered while lamenting the uselessness of everything: the possibility of eternal life after this one.
"Both go to the same place" - Eccl. 3:20a
Of course Qoheleth is going to question the justice of God when he sees bad things happening to good people and good things happening to bad people in this life. Jesus, in the Beatitudes, flips the prophetic expectation of justice in this life on its head by saying, for instance, "Blessed are the poor in spirit, for theirs is the kingdom of heaven" (Matt. 5:3). Instead of articulating that justice will be completely served in this life (which Qoheleth rightly notes is not, does not, and has not happened), Jesus there offers a second chance, taking all of eternity, for justice to be served properly.
As to a Christian equivalent for Sukkot, there isn't a universal one, but many Orthodox Christians draw parallel between Sukkot and the Feast of the Transfiguration. Catholics and many other Christians celebrate this feast as well, normally in August. The parallel is drawn by considering the fact that at the Transfiguration, when Moses and Elijah are said to have appeared at Jesus's left and right, Peter asked if he, James and John should pitch three tents (i.e. tabernacles, i.e. sukkot) for the three transfigured beings. The commemoration of the holiday outside of the liturgy, however, is not held in parallel.